This course treats a function as a rule that gives every input exactly one output, and then studies the functions you will use throughout higher mathematics. You will sketch parabolas from their vertex and roots and use them to solve quadratic inequations. You will work with logarithms as a function, their graphs and their sign rules. You will solve equations and inequalities with the modulus by the critical point method, find domains of square root expressions, and work with the greatest integer and fractional part functions. The course ends with a table of the standard graphs, domains and ranges.
Middle Years
Functions and Graphs
4-week course.
- 16 lessons
Overview
Lessons
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What is a function
A function sends each element of one set to exactly one element of another. Arrow diagrams, image and pre-image, and the domain, codomain and range of a function.
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Polynomial, constant, identity and linear functions
Polynomial functions and their degree, the rule for a domain that is not given, and the graphs of constant, identity and linear functions.
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Quadratic functions and their roots
The quadratic formula, the discriminant and the nature of roots, the sum, product and difference of the roots, and the factored form a(x − α)(x − β).
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Graph of a quadratic function
Vertex form, upward and downward parabolas, maximum and minimum values, ranges, and step by step sketches of parabolas.
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Quadratic inequations
Solving ax² + bx + c < 0, ≤ 0, > 0 and ≥ 0 by reading the sign of the parabola, and the short rule: less than zero, x lies between the roots.
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Logarithms: definition and laws
Logarithm as an exponent, why the base must be positive and not 1, and the twelve laws of logarithms with worked evaluations.
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Graphs of logarithms and log inequalities
The graphs of log to base 2 and base ½, the two shapes for a > 1 and 0 < a < 1, the sign rules for a logarithm, and simple logarithmic inequalities.
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Solving problems on logarithms
Logarithmic equations and standard simplifications using the product, power, chain and change of base laws, always checking answers against the conditions.
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The modulus function
The modulus |x|, its graph, its domain and range, the properties used in every modulus problem, and the rules for |x| < a and |x| > a.
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Modulus equations and inequalities
Solving |x| = k and |x| = |y|, removing the modulus sign once the sign is known, and solving equations and inequalities with one modulus.
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Modulus problems by critical points
The critical point method: cut the number line where each modulus changes form, solve on each piece and keep only the common values.
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Harder modulus problems
Equations solved by the equality case of the triangle inequality, a modulus inside a modulus, inequalities with fractions, and counting integer solutions.
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The square root function and domains
The square root function and its graph, and finding the domain of expressions with square roots by solving an inequality.
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The greatest integer function
The step of a real number [x], values for negative numbers, the staircase graph, its properties, and inequalities and domains with [x].
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The fractional part function
The fractional part x − [x], its values for negative numbers, its sawtooth graph with period 1, and equations mixing [x] and the fractional part.
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Standard graphs, domains and ranges
Exponential functions, 1/x, 1/x², x², x³ and x + 1/x, and a summary table of the domains, ranges and graphs of every function in the chapter.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.
Class notes
Functions and Graphs
Typed from the teacher's handwritten class slides, 41 pages.
Download the notes (PDF, 1.3 MB)